I have only two on my computer: Ukrainian Unicode and English International and use it for Russian, Ukrainian, English, French and Irish. I can type Russian specific letters using Right ALT and і, є, ї keys, accented Latin characters: using ' key and the key of letter I need. And that's it :)

Do you use Linux? Because when I did, it worked exactly like this for me, which was fantastic (I could even type almost all Polish characters, except ł, but that's easily fixable), but that computer was broken and now I use Windows 8 on a new one. Here the English international layout has far more limitations, and the Ukrainian one allows only typing of our language's characters, not of Belarusian or Russian.

Yes, I am. However, I have been using English International on Windows 7 some years ago and it was totally fine, I even could type French letter ç which I unfortunately can't now. I don't know how it changed in Windows 8. As to Ukrainian Unicode you also can have it on Windows, you just need to install it separately.
Try this link: http://r2u.org.ua/data/other/uk_unicode_keyboard_winxp.zip

Yesterday I did the first two lessons of the first skill and when I had to type in Ukrainian, I typed in it with Latin characters. For example, I wrote dim (=house), tato (=dad), etc. and my answers were accepted as correct. So I don't understand your problem. Just type in Ukrainian with Latin characters, as I do! :)

I didn't find a latin character accepted for ц. If there is one, I'm interested to know.

It's probably a bad habit to write/read it with latin character. And a bad habit is generally harder to loose than it would have been to take the good habit from the beginning.

My real problem is that with the Ukrainian keyboard that I use, the shortcuts I normally use (on my latin keyboard) for "undo" and "select all" (I use them all the time) do respectively close the tab and close the application...
I just keep involuntarily closing the lesson I'm doing all the time... this is quite annoying.

I'll try to use Cyrillic. Just hope to not automatically hit to much my (French) "cmd+z" and "cmd+a" which, on Ukrainian and English keyboards respectively close the tab and close the application. After happening twice in less than 5min, I went out before throwing my computer across the room. ;)

I'm especially looking forward to the eventual app release for Ukrainian for this specific reason. At least with other languages in the app, the app always automatically switches my keyboard to the language I'm supposed to type in for each sentence it gives. It's not enough of an issue to make me quit learning Ukrainian, but it is a bit distracting from learning when I'm messing with technical things like switching keyboards. I wonder if there's a way to have the desktop version similarly auto-switch your keyboards for you? Not bothered though, just happily looking forward to the eventual app release!

Someone in one of the Russian threads shared this firefox add-on http://www.benya.com/transliterator/ IMO, it's the easiest thing to use. You just hit F2 to turn it on or off. Since it's part of your browser, it will only work on websites of course, not when typing in word or something like that. I have other cyrillic keyboards installed, but this one is the easiest for beginners, IMO. It transliterates from Roman to cyrillic which is exactly what you want. And it supports the Ukrainian language. This is good for people who are just fooling around and want to try Russian or Ukrainian. You can get a more standard keyboard later.

What shortcuts would be good for that? I now use caps lock to switch between usaltgr and slavonic multilingual, which so far can type all languages for English speakers, but maybe I should start using Windows + function keys when I add a third one.

shift+menu (where menu is a key I have between my right alt and ctrl keys) toggles Japanese input on and off. I use IBus for Japanese input (and in fact I also have it set up to cater to LaTeX input for when I need to use a maths symbol)

I use win+Fn where win between my left alt and ctrl keys and Fn is a function key to swap between actual keyboard layouts. I use F1 for Dvorak Standard, F2 for Programmer's Dvorak, and F8 for QWERTY (some games I run don't support the Dvorak layout well at all). I have F3 set up for LH Dvorak and F4 for RH (but I rarely use them) and F5 for a Ukrainian keyboard should I ever get around to doing the course.

A standard way of cycling through keyboard layouts is shift+space, but it's not something I use myself and IMO only works well if you're cycling between two.

Yeah, it's kind of a pain, but the nice part of the constant codex switching is that your brain can also "switch" back and forth between English and Ukrainian. Plus, learning to type in another keyboard is exciting and interesting at the same time

i only did a lesson or two of ukrainian, but i just typed the english equivalent letters and it accepted everything just fine. for example, for "дім" i just typed "dim". i didn't need to get any special ukrainian character keyboard.

This is what I do. I use the keyboards for English and Ukrainian and just use command+space to switch between the two. I have the keyboard viewer open and tucked out of the way so I can see what I'm typing. Not that difficult on a Mac.

I use Command + double space, because if I press space only one time it opens the Spotlight, and it's very convenient
I have Ukrainian phonetic keyboard installed and my own mix of Polish and Spanish (Polish layout with posibility to make both ą ę ć ż ź and ¿, ¡, á, é, ú, etc)

If you want to learn a language which uses a different alphabet, that also means learning the new alphabet. You'll have to sooner or later, and believe me, sooner is better. Relying on Latin letters will just slow you down in the long term.